Long before the very first computer was invented, people have been predicting the weather using crude methods like observing the sky, such as recurring astronomical and meteorological events, and nature in general. One of the earliest civilization which tried to predict short-term weather changes were the Babylonians around 650 BC, they based their prediction on the appearance of clouds and optical phenomena such as halos.
Chinese astronomers developed a calendar that divided the year into 24 festivals, each festival associated with a different type of weather. Aristotle, the famous Greek philosopher wrote a treatise that included theories about the formation of rain, clouds, hail, wind, thunder, lightning, and hurricanes. He made some remarkable observations concerning the weather, and for many years was the authority on weather theory, only to be overthrown as erroneous in the 17th century.
By the end of the Renaissance, it had become increasingly evident that the speculations of the natural philosophers were inadequate and that greater knowledge was necessary to further our understanding of the atmosphere. Instruments were invented to measure the properties of the atmosphere, such as moisture, temperature, and pressure. The invention of the telegraph in the 1860s allowed the routine transmission of weather observations to and from observers and compilers. Using these data, crude weather maps were drawn and surface wind patterns and storm systems could be identified and studied. Weather-observing stations began appearing all across the globe, eventually spawning the birth of synoptic weather forecasting.
Modern weather forecasting involves a combination of computer models, observation, and a knowledge of trends and patterns. Using these methods, reasonably accurate forecasts can be made up to about five days in advance. Beyond that, detailed forecasts are less useful, since atmospheric conditions such as temperature and wind direction are very complex.
But you don’t need to have a supercomputer or weather balloon to try your hand at forecasting, though. The most basic weather forecasting consists of simple observation. For example, you can look up at the clouds and try to recognize telltale patterns as people did in the past. High, wispy clouds usually presage good weather. An overcast sky means rain or snow is on the way. Certain weather features seem to be associated with certain types of weather, at least most of the time.